This invention relates to the field of tiles and tilings. The field includes the familiar floor and kitchen-counter top tiles and tilings of commerce and their like, but also extends to the sometimes more abstract areas of art, design, and mathematics.
1. Some Definitions
I have adapted a few of the notions and definitions of the mathematics of tiles and tilings as follows. A tile is a two-dimensional closed shape which fits together edge-to-edge with other different or similar two-dimensional shapes, as do jig-saw puzzle pieces or bricks, to cover a flat surface of indefinite extent. Such a covering is called a tiling if it has no gap between tiles nor any overlap of one tile on another. Adding thickness to a two dimensional tile will make it a three dimensional object which is also called a tile. A tile or a set of tiles is said to "tile the plane" if indefinitely large numbers of duplicates of the tile or of the members of the set of tiles can fit together without gap or lap in a tiling. The term "the plane" refers to the flat indefinitely extensive plane of Euclidian geometry. As a verb, "tile" means to form a tiling.
A figurative tile is one whose shape is the recognizable outline, or figure, of a person or an animal. A figurative tiling is a tiling composed of such figurative tiles. Examples of such figurative tiles and tilings are to be found in the work of the late Dutch artist, M. C. Escher.
A variably assemblable tiling is one whose tiles function so as to fit together or to interlock with one another in a variety of different ways, allowing a plurality of different tilings to be made. Perhaps the simplest tile to form such a plurality is the common brick with its many different arrangements and patternings in walls and pavings. Sets of curved sided tiles that are variably assemblable are somewhat more difficult to design, as may be seen in U.S. Pat. No. 4,217,740 of Aug. 19, 1980 to Assanti.
2. Prior Art
U.S. Pat. No. 4,133,152 of Jan. 9, 1979 to Penrose shows a crudely figurative and variably assemblable set of two tiles that has since become known as "Penrose's chickens". This is the only known variably assemblable figurative tile set that is not the work of the present inventor, John A. L. Osborn.
Though Penrose's underlying geometry is of seminal importance, these "chicken" shapes have many disadvantages both when it comes to forming aesthetically pleasing tilings and when the tiles are to act as pieces in a puzzle.
(a) There are only two different shapes in this "chicken" set. Thus there is little variety in the pieces to interest the eye, and the options for tile choice in assembling a puzzle are severely limited. PA1 (b) The two pieces can fit together in only four ways,which is such a low number of options that it adds virtually nothing to the perceived complexity of a puzzle. Nor does it add significantly to the visual interest of a tiling used for aesthetic effect. PA1 (c) The "chicken" shapes almost entirely lack the emblematic quality of clear recognizability that a figurative tile outline should have in order to convey the significance of its shape to the viewer even when it is surrounded by a swirl of differently oriented similar shapes as it is in a tiling. When viewing a patch of "chicken" shapes one's eye tends to see only lumps and bumps. The geometric shapes they derive from are much more attractive and interesting. PA1 (d) For use as puzzle pieces the non-interlocking quality of the "chicken" shapes makes them easily jostled into disorder, and hard to move and save for a later session of play. PA1 (e) In working with the "chicken" tiles, it does not quickly become apparent that there are many possible tilings. Certain combinations of a few tiles occur again and again, and one has to assemble a very large number of tiles before it becomes apparent that indeed the larger patterns do not repeat periodically. Of recent years the Penrose non-repetitive tilings, including the Penrose chickens, have come to be called "quasiperiodic", a name that calls attention to the fact that they give the impression of being periodic, or repetitive. PA1 (a) There are at least ten distinct subsets of my eight-beetle set which will tile the plane. Five of these are capable of doing so in an infinite number of different ways. The full eight member set is one of this latter group of five subsets. PA1 (b) Any single one of my beetle tiles will fit together one-on-one with the other beetle tiles, including a duplicate of itself, in a total of 72 distinct ways. This provides a great variety of choice in assembling a tiling, as, for instance, in doing a puzzle where a defined border is to be filled in. It also allows the creation of interestingly shaped and aesthetically pleasing patches of tiles. Some of these patches of tiles may be extensible indefinitely, while others may have self-defined geometric outlines by virtue of the development of untilable borders. PA1 (c) Each of the beetle tiles has a family resemblance to the others in the eight member set, but their six-legged shapes are quite distinct while also being emblematically insect-like even without the embellishment of internal drawing. Even when surrounded by other beetle shapes each beetle is distinct. PA1 (d) My beetle tiles interlock in such a way that a tiled patch of puzzle pieces can be slid around on a table surface quite freely without disturbing the arrangement of the tiles. PA1 (e) People working or playing with my beetle tiles become aware almost immediately that there are a variety of ways to tile a surface with them, even though it takes a while to realize the full extent of that variety.